Quantile of the normal distribution
For a quantile the probability is given and the value is wanted: work back through z.
Worked example
The fill volume is normal with μ = 100 ml and σ = 20 ml. Which volume do 97.5% of packs fall below? (z = 1.960)
Worked solution
- 1.x = μ + z · σ = 100 + 1.960 · 20 = 139.20
Typical mistake: Confusing z with the probability p and calculating p·σ.
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
The fill volume is normal with μ = 100 ml and σ = 5 ml. Which volume do 97.5% of packs fall below? (z = 1.960)
x (2 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- z = (x − μ) / σ
- P(X ≤ a) = Φ(z)
- P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
- Φ(−z) = 1 − Φ(z); quantile: x = μ + z · σ
Typical mistake: Confusing z with the probability p and calculating p·σ.
Frequently asked questions
What is the typical mistake in "Quantile of the normal distribution"?
Confusing z with the probability p and calculating p·σ.
How do I read negative z values?
Tables usually list positive z only: Φ(−z) = 1 − Φ(z).